1492:
1351:
885:
1236:
1862:
472:
1132:
The geometric relation between projective twistor space and complexified compactified
Minkowski space is the same as the relation between lines and two-planes in twistor space; more precisely, twistor space is
609:
105:
1384:
974:
776:
663:
1594:
1174:
1780:
1725:
1643:
1095:
1034:
712:
347:
237:
198:
303:
266:
166:
1127:
534:
1002:
2013:
1804:
1619:
1539:
1517:
1376:
1258:
381:
502:
1266:
1063:
1189:
239:
parametrizes such isomorphisms together with complex coordinates. Thus one complex coordinate describes the identification and the other two describe a point in
1667:
787:
1809:
2006:
139:: "the shortest path between two truths in the real domain passes through the complex domain." Therefore when studying four-dimensional space
1981:
1933:
979:
This incidence relation is preserved under an overall re-scaling of the twistor, so usually one works in projective twistor space, denoted
1999:
1962:
1487:{\displaystyle \mathbb {M} =F_{2}(\mathbb {T} )=\operatorname {Gr} _{2}(\mathbb {C} ^{4})=\operatorname {Gr} _{2,4}(\mathbb {C} )}
389:
1065:
it is related to a line in projective twistor space where we can see the incidence relation as giving the linear embedding of a
43:
714:
the indexes on the matrices. This twistor space is a four-dimensional complex vector space, whose points are denoted by
543:
717:
614:
1547:
1923:
1139:
910:
2147:
1730:
314:
205:
2043:
1688:
2121:
1628:
2116:
1950:
891:
1068:
1007:
672:
320:
210:
171:
1674:
279:
242:
142:
2064:
1865:
273:
33:
982:
21:
1787:
1602:
1522:
1500:
1359:
1241:
364:
1678:
1346:{\displaystyle \mathbb {P} =F_{1}(\mathbb {T} )=\mathbb {CP} ^{3}=\mathbf {P} (\mathbb {C} ^{4})}
902:
480:
124:
1100:
507:
115:, twistor space is useful for conceptualizing the way photons travel through space, using four
2091:
1977:
1958:
1929:
1682:
1904:
1622:
1042:
136:
1231:{\displaystyle \mathbb {P} \xleftarrow {\mu } \mathbb {F} \xrightarrow {\nu } \mathbb {M} }
2059:
895:
358:
2126:
2022:
1652:
779:
666:
116:
37:
25:
1891:
880:{\displaystyle \Sigma (Z)=\omega ^{A}{\bar {\pi }}_{A}+{\bar {\omega }}^{A'}\pi _{A'}}
2141:
1908:
1869:
1670:
1184:
269:
112:
108:
204:
respecting orientation and metric between the two are considered. It turns out that
1646:
901:
Points in
Minkowski space are related to subspaces of twistor space through the
537:
201:
17:
1180:
310:
306:
120:
1991:
1857:{\displaystyle \mathbf {P} _{5}\cong \mathbf {P} (\wedge ^{2}\mathbb {T} )}
1892:"Twistor theory: An approach to the quantisation of fields and space-time"
1217:
1202:
200:
However, since there is no canonical way of doing so, instead all
119:. He also posits that twistor space may aid in understanding the
1995:
687:
467:{\displaystyle \Omega ^{A}(x)=\omega ^{A}-ix^{AA'}\pi _{A'}}
383:, the solutions to the twistor equation are of the form
1812:
1790:
1733:
1691:
1655:
1631:
1605:
1550:
1525:
1503:
1387:
1362:
1269:
1244:
1192:
1142:
1103:
1071:
1045:
1010:
985:
913:
790:
720:
675:
617:
546:
510:
483:
392:
367:
323:
282:
245:
213:
174:
145:
100:{\displaystyle \nabla _{A'}^{(A}\Omega _{^{}}^{B)}=0}
46:
604:{\displaystyle x^{AA'}=\sigma _{\mu }^{AA'}x^{\mu }}
2109:
2084:
2077:
2052:
2036:
2029:
771:{\displaystyle Z^{\alpha }=(\omega ^{A},\pi _{A'})}
658:{\displaystyle \sigma _{\mu }=(I,{\vec {\sigma }})}
1890:
1856:
1798:
1774:
1719:
1661:
1637:
1613:
1589:{\displaystyle \mathbb {F} =F_{1,2}(\mathbb {T} )}
1588:
1533:
1511:
1486:
1370:
1345:
1252:
1230:
1168:
1121:
1089:
1057:
1028:
996:
968:
879:
770:
706:
657:
603:
528:
496:
466:
375:
341:
297:
260:
231:
192:
160:
99:
1378:is the compactified complexified Minkowski space
1889:Penrose, R.; MacCallum, M.A.H. (February 1973).
1169:{\displaystyle \mathbb {T} :=\mathbb {C} ^{4}.}
1004:, which is isomorphic as a complex manifold to
1784:The compactified complexified Minkowski space
969:{\displaystyle \omega ^{A}=ix^{AA'}\pi _{A'}.}
2007:
8:
898:C(1,3) of compactified Minkowski spacetime.
1775:{\displaystyle c^{-1}=\mu \circ \nu ^{-1}.}
2081:
2033:
2014:
2000:
1992:
1847:
1846:
1840:
1828:
1819:
1814:
1811:
1792:
1791:
1789:
1760:
1738:
1732:
1708:
1690:
1654:
1630:
1606:
1604:
1579:
1578:
1563:
1552:
1551:
1549:
1527:
1526:
1524:
1505:
1504:
1502:
1477:
1476:
1458:
1442:
1438:
1437:
1424:
1410:
1409:
1400:
1389:
1388:
1386:
1364:
1363:
1361:
1334:
1330:
1329:
1320:
1311:
1307:
1304:
1303:
1292:
1291:
1282:
1271:
1270:
1268:
1246:
1245:
1243:
1224:
1223:
1209:
1208:
1194:
1193:
1191:
1157:
1153:
1152:
1144:
1143:
1141:
1108:
1102:
1081:
1077:
1074:
1073:
1070:
1044:
1020:
1016:
1013:
1012:
1009:
990:
987:
986:
984:
952:
934:
918:
912:
866:
851:
840:
839:
829:
818:
817:
810:
789:
754:
741:
725:
719:
686:
674:
641:
640:
622:
616:
595:
577:
572:
551:
545:
515:
509:
488:
482:
453:
435:
419:
397:
391:
369:
368:
366:
333:
329:
326:
325:
322:
289:
285:
284:
281:
252:
248:
247:
244:
223:
219:
216:
215:
212:
181:
177:
176:
173:
168:it might be valuable to identify it with
152:
148:
147:
144:
82:
78:
74:
61:
51:
45:
1881:
1925:One to Nine: The Inner Life of Numbers
1720:{\displaystyle c=\nu \circ \mu ^{-1}}
1497:and the correspondence space between
7:
111:and Malcolm MacCallum. According to
1638:{\displaystyle \operatorname {Gr} }
1179:It has associated to it the double
611:is a point in Minkowski space. The
107:. It was described in the 1960s by
894:which is a quadruple cover of the
791:
394:
71:
48:
14:
1974:An introduction to twistor theory
1972:Huggett, S.A.; Tod, K.P. (1994).
1955:Twistor Geometry and Field Theory
1928:. Doubleday Canada. p. 142.
1090:{\displaystyle \mathbb {CP} ^{1}}
1029:{\displaystyle \mathbb {CP} ^{3}}
707:{\displaystyle A,A^{\prime }=1,2}
342:{\displaystyle \mathbb {CP} ^{3}}
232:{\displaystyle \mathbb {CP} ^{3}}
193:{\displaystyle \mathbb {C} ^{2}.}
1829:
1815:
1607:
1321:
1260:is the projective twistor space
298:{\displaystyle \mathbb {R} ^{4}}
261:{\displaystyle \mathbb {R} ^{4}}
161:{\displaystyle \mathbb {R} ^{4}}
1976:. Cambridge University Press.
1957:. Cambridge University Press.
1851:
1833:
1583:
1575:
1481:
1473:
1448:
1433:
1414:
1406:
1340:
1325:
1296:
1288:
845:
823:
800:
794:
765:
734:
652:
646:
631:
409:
403:
317:on complex projective 3-space
86:
62:
1:
997:{\displaystyle \mathbb {PT} }
890:which is invariant under the
135:In the (translated) words of
1909:10.1016/0370-1573(73)90008-2
1799:{\displaystyle \mathbb {M} }
1614:{\displaystyle \mathbf {P} }
1534:{\displaystyle \mathbb {M} }
1512:{\displaystyle \mathbb {P} }
1371:{\displaystyle \mathbb {M} }
1253:{\displaystyle \mathbb {P} }
376:{\displaystyle \mathbb {M} }
497:{\displaystyle \omega ^{A}}
2164:
315:holomorphic vector bundles
206:complex projective 3-space
1122:{\displaystyle \pi _{A'}}
529:{\displaystyle \pi _{A'}}
2023:Topics of twistor theory
2044:Background independence
1922:Hodges, Andrew (2010).
2122:Twistor correspondence
1858:
1800:
1776:
1721:
1663:
1639:
1615:
1590:
1535:
1513:
1488:
1372:
1347:
1254:
1232:
1170:
1123:
1091:
1059:
1058:{\displaystyle x\in M}
1030:
998:
970:
881:
772:
708:
659:
605:
530:
498:
468:
377:
343:
311:correspond bijectively
299:
262:
233:
194:
162:
101:
2117:Twistor string theory
2078:Mathematical concepts
1859:
1801:
1777:
1722:
1664:
1640:
1616:
1591:
1536:
1514:
1489:
1373:
1348:
1255:
1233:
1171:
1124:
1092:
1060:
1031:
999:
971:
882:
773:
709:
660:
606:
531:
499:
469:
378:
344:
300:
274:self-dual connections
263:
234:
195:
163:
102:
2065:Theory of everything
1810:
1788:
1731:
1689:
1653:
1629:
1603:
1548:
1523:
1501:
1385:
1360:
1267:
1242:
1190:
1140:
1101:
1069:
1043:
1008:
983:
911:
788:
718:
673:
615:
544:
508:
481:
390:
365:
321:
280:
268:. It turns out that
243:
211:
172:
143:
44:
36:of solutions of the
34:complex vector space
1868:; the image is the
1221:
1206:
590:
131:Informal motivation
90:
69:
22:theoretical physics
1854:
1796:
1772:
1717:
1677:gives rise to two
1659:
1635:
1611:
1586:
1531:
1509:
1484:
1368:
1343:
1250:
1228:
1166:
1119:
1087:
1055:
1026:
994:
966:
903:incidence relation
877:
768:
704:
655:
601:
568:
526:
494:
464:
373:
339:
295:
258:
229:
190:
158:
125:weak nuclear force
97:
70:
47:
2148:Complex manifolds
2135:
2134:
2110:Physical concepts
2105:
2104:
2092:Penrose transform
2073:
2072:
1983:978-0-521-45689-0
1935:978-0-385-67266-5
1866:Plücker embedding
1683:Penrose transform
1662:{\displaystyle F}
1222:
1207:
848:
826:
649:
536:are two constant
353:Formal definition
2155:
2082:
2034:
2016:
2009:
2002:
1993:
1987:
1968:
1940:
1939:
1919:
1913:
1912:
1894:
1886:
1863:
1861:
1860:
1855:
1850:
1845:
1844:
1832:
1824:
1823:
1818:
1805:
1803:
1802:
1797:
1795:
1781:
1779:
1778:
1773:
1768:
1767:
1746:
1745:
1726:
1724:
1723:
1718:
1716:
1715:
1675:double fibration
1668:
1666:
1665:
1660:
1644:
1642:
1641:
1636:
1623:projective space
1620:
1618:
1617:
1612:
1610:
1595:
1593:
1592:
1587:
1582:
1574:
1573:
1555:
1540:
1538:
1537:
1532:
1530:
1518:
1516:
1515:
1510:
1508:
1493:
1491:
1490:
1485:
1480:
1469:
1468:
1447:
1446:
1441:
1429:
1428:
1413:
1405:
1404:
1392:
1377:
1375:
1374:
1369:
1367:
1352:
1350:
1349:
1344:
1339:
1338:
1333:
1324:
1316:
1315:
1310:
1295:
1287:
1286:
1274:
1259:
1257:
1256:
1251:
1249:
1237:
1235:
1234:
1229:
1227:
1213:
1212:
1198:
1197:
1175:
1173:
1172:
1167:
1162:
1161:
1156:
1147:
1128:
1126:
1125:
1120:
1118:
1117:
1116:
1097:parametrized by
1096:
1094:
1093:
1088:
1086:
1085:
1080:
1064:
1062:
1061:
1056:
1035:
1033:
1032:
1027:
1025:
1024:
1019:
1003:
1001:
1000:
995:
993:
975:
973:
972:
967:
962:
961:
960:
947:
946:
945:
923:
922:
886:
884:
883:
878:
876:
875:
874:
861:
860:
859:
850:
849:
841:
834:
833:
828:
827:
819:
815:
814:
777:
775:
774:
769:
764:
763:
762:
746:
745:
730:
729:
713:
711:
710:
705:
691:
690:
664:
662:
661:
656:
651:
650:
642:
627:
626:
610:
608:
607:
602:
600:
599:
589:
588:
576:
564:
563:
562:
535:
533:
532:
527:
525:
524:
523:
503:
501:
500:
495:
493:
492:
473:
471:
470:
465:
463:
462:
461:
448:
447:
446:
424:
423:
402:
401:
382:
380:
379:
374:
372:
348:
346:
345:
340:
338:
337:
332:
304:
302:
301:
296:
294:
293:
288:
267:
265:
264:
259:
257:
256:
251:
238:
236:
235:
230:
228:
227:
222:
199:
197:
196:
191:
186:
185:
180:
167:
165:
164:
159:
157:
156:
151:
137:Jacques Hadamard
106:
104:
103:
98:
89:
81:
80:
79:
68:
60:
59:
2163:
2162:
2158:
2157:
2156:
2154:
2153:
2152:
2138:
2137:
2136:
2131:
2101:
2069:
2060:Quantum gravity
2053:Final objective
2048:
2025:
2020:
1990:
1984:
1971:
1965:
1948:
1944:
1943:
1936:
1921:
1920:
1916:
1897:Physics Reports
1888:
1887:
1883:
1878:
1836:
1813:
1808:
1807:
1806:is embedded in
1786:
1785:
1756:
1734:
1729:
1728:
1704:
1687:
1686:
1679:correspondences
1651:
1650:
1627:
1626:
1601:
1600:
1559:
1546:
1545:
1521:
1520:
1499:
1498:
1454:
1436:
1420:
1396:
1383:
1382:
1358:
1357:
1328:
1302:
1278:
1265:
1264:
1240:
1239:
1188:
1187:
1151:
1138:
1137:
1109:
1104:
1099:
1098:
1072:
1067:
1066:
1041:
1040:
1011:
1006:
1005:
981:
980:
953:
948:
938:
930:
914:
909:
908:
896:conformal group
867:
862:
852:
838:
816:
806:
786:
785:
755:
750:
737:
721:
716:
715:
682:
671:
670:
618:
613:
612:
591:
581:
555:
547:
542:
541:
516:
511:
506:
505:
484:
479:
478:
454:
449:
439:
431:
415:
393:
388:
387:
363:
362:
359:Minkowski space
355:
324:
319:
318:
283:
278:
277:
246:
241:
240:
214:
209:
208:
175:
170:
169:
146:
141:
140:
133:
117:complex numbers
75:
52:
42:
41:
12:
11:
5:
2161:
2159:
2151:
2150:
2140:
2139:
2133:
2132:
2130:
2129:
2127:Twistor theory
2124:
2119:
2113:
2111:
2107:
2106:
2103:
2102:
2100:
2099:
2094:
2088:
2086:
2079:
2075:
2074:
2071:
2070:
2068:
2067:
2062:
2056:
2054:
2050:
2049:
2047:
2046:
2040:
2038:
2031:
2027:
2026:
2021:
2019:
2018:
2011:
2004:
1996:
1989:
1988:
1982:
1969:
1963:
1945:
1942:
1941:
1934:
1914:
1903:(4): 241–315.
1880:
1879:
1877:
1874:
1853:
1849:
1843:
1839:
1835:
1831:
1827:
1822:
1817:
1794:
1771:
1766:
1763:
1759:
1755:
1752:
1749:
1744:
1741:
1737:
1714:
1711:
1707:
1703:
1700:
1697:
1694:
1658:
1634:
1609:
1599:In the above,
1597:
1596:
1585:
1581:
1577:
1572:
1569:
1566:
1562:
1558:
1554:
1529:
1507:
1495:
1494:
1483:
1479:
1475:
1472:
1467:
1464:
1461:
1457:
1453:
1450:
1445:
1440:
1435:
1432:
1427:
1423:
1419:
1416:
1412:
1408:
1403:
1399:
1395:
1391:
1366:
1354:
1353:
1342:
1337:
1332:
1327:
1323:
1319:
1314:
1309:
1306:
1301:
1298:
1294:
1290:
1285:
1281:
1277:
1273:
1248:
1226:
1220:
1216:
1211:
1205:
1201:
1196:
1185:flag manifolds
1177:
1176:
1165:
1160:
1155:
1150:
1146:
1115:
1112:
1107:
1084:
1079:
1076:
1054:
1051:
1048:
1039:Given a point
1023:
1018:
1015:
992:
989:
977:
976:
965:
959:
956:
951:
944:
941:
937:
933:
929:
926:
921:
917:
888:
887:
873:
870:
865:
858:
855:
847:
844:
837:
832:
825:
822:
813:
809:
805:
802:
799:
796:
793:
780:hermitian form
767:
761:
758:
753:
749:
744:
740:
736:
733:
728:
724:
703:
700:
697:
694:
689:
685:
681:
678:
667:Pauli matrices
654:
648:
645:
639:
636:
633:
630:
625:
621:
598:
594:
587:
584:
580:
575:
571:
567:
561:
558:
554:
550:
522:
519:
514:
491:
487:
475:
474:
460:
457:
452:
445:
442:
438:
434:
430:
427:
422:
418:
414:
411:
408:
405:
400:
396:
371:
354:
351:
336:
331:
328:
292:
287:
270:vector bundles
255:
250:
226:
221:
218:
189:
184:
179:
155:
150:
132:
129:
96:
93:
88:
85:
77:
73:
67:
64:
58:
55:
50:
26:twistor theory
13:
10:
9:
6:
4:
3:
2:
2160:
2149:
2146:
2145:
2143:
2128:
2125:
2123:
2120:
2118:
2115:
2114:
2112:
2108:
2098:
2097:Twistor space
2095:
2093:
2090:
2089:
2087:
2083:
2080:
2076:
2066:
2063:
2061:
2058:
2057:
2055:
2051:
2045:
2042:
2041:
2039:
2035:
2032:
2028:
2024:
2017:
2012:
2010:
2005:
2003:
1998:
1997:
1994:
1985:
1979:
1975:
1970:
1966:
1964:0-521-42268-X
1960:
1956:
1952:
1947:
1946:
1937:
1931:
1927:
1926:
1918:
1915:
1910:
1906:
1902:
1898:
1893:
1885:
1882:
1875:
1873:
1871:
1870:Klein quadric
1867:
1841:
1837:
1825:
1820:
1782:
1769:
1764:
1761:
1757:
1753:
1750:
1747:
1742:
1739:
1735:
1712:
1709:
1705:
1701:
1698:
1695:
1692:
1684:
1680:
1676:
1672:
1671:flag manifold
1656:
1648:
1632:
1624:
1570:
1567:
1564:
1560:
1556:
1544:
1543:
1542:
1470:
1465:
1462:
1459:
1455:
1451:
1443:
1430:
1425:
1421:
1417:
1401:
1397:
1393:
1381:
1380:
1379:
1335:
1317:
1312:
1299:
1283:
1279:
1275:
1263:
1262:
1261:
1218:
1214:
1203:
1199:
1186:
1182:
1163:
1158:
1148:
1136:
1135:
1134:
1130:
1113:
1110:
1105:
1082:
1052:
1049:
1046:
1037:
1021:
963:
957:
954:
949:
942:
939:
935:
931:
927:
924:
919:
915:
907:
906:
905:
904:
899:
897:
893:
892:group SU(2,2)
871:
868:
863:
856:
853:
842:
835:
830:
820:
811:
807:
803:
797:
784:
783:
782:
781:
778:, and with a
759:
756:
751:
747:
742:
738:
731:
726:
722:
701:
698:
695:
692:
683:
679:
676:
668:
643:
637:
634:
628:
623:
619:
596:
592:
585:
582:
578:
573:
569:
565:
559:
556:
552:
548:
539:
520:
517:
512:
489:
485:
458:
455:
450:
443:
440:
436:
432:
428:
425:
420:
416:
412:
406:
398:
386:
385:
384:
360:
352:
350:
334:
316:
312:
308:
290:
275:
271:
253:
224:
207:
203:
187:
182:
153:
138:
130:
128:
126:
122:
118:
114:
113:Andrew Hodges
110:
109:Roger Penrose
94:
91:
83:
76:
65:
56:
53:
39:
35:
31:
30:twistor space
27:
23:
19:
2096:
1973:
1954:
1949:Ward, R.S.;
1924:
1917:
1900:
1896:
1884:
1783:
1647:Grassmannian
1598:
1496:
1355:
1178:
1131:
1038:
978:
900:
889:
538:Weyl spinors
476:
356:
202:isomorphisms
134:
29:
24:(especially
15:
1951:Wells, R.O.
1621:stands for
18:mathematics
2037:Principles
2030:Objectives
1876:References
1681:(see also
361:, denoted
307:instantons
1838:∧
1826:≅
1762:−
1758:ν
1754:∘
1751:μ
1740:−
1710:−
1706:μ
1702:∘
1699:ν
1471:
1431:
1219:ν
1204:μ
1181:fibration
1106:π
1050:∈
950:π
916:ω
864:π
846:¯
843:ω
824:¯
821:π
808:ω
792:Σ
752:π
739:ω
727:α
688:′
647:→
644:σ
624:μ
620:σ
597:μ
574:μ
570:σ
513:π
486:ω
451:π
426:−
417:ω
395:Ω
121:asymmetry
72:Ω
49:∇
40:equation
2142:Category
2085:Twistors
1953:(1991).
1215:→
1200:←
1114:′
958:′
943:′
872:′
857:′
760:′
665:are the
586:′
560:′
521:′
459:′
444:′
57:′
1864:by the
669:, with
123:of the
38:twistor
32:is the
1980:
1961:
1932:
1673:. The
1649:, and
1238:where
477:where
272:with
1978:ISBN
1959:ISBN
1930:ISBN
1727:and
1519:and
1356:and
540:and
504:and
357:For
20:and
1905:doi
1685:),
1541:is
1183:of
313:to
276:on
28:),
16:In
2144::
1899:.
1895:.
1872:.
1669:a
1645:a
1633:Gr
1625:,
1456:Gr
1422:Gr
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1129:.
1036:.
349:.
309:)
127:.
2015:e
2008:t
2001:v
1986:.
1967:.
1938:.
1911:.
1907::
1901:6
1852:)
1848:T
1842:2
1834:(
1830:P
1821:5
1816:P
1793:M
1770:.
1765:1
1748:=
1743:1
1736:c
1713:1
1696:=
1693:c
1657:F
1608:P
1584:)
1580:T
1576:(
1571:2
1568:,
1565:1
1561:F
1557:=
1553:F
1528:M
1506:P
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1478:C
1474:(
1466:4
1463:,
1460:2
1452:=
1449:)
1444:4
1439:C
1434:(
1426:2
1418:=
1415:)
1411:T
1407:(
1402:2
1398:F
1394:=
1390:M
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1336:4
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1326:(
1322:P
1318:=
1313:3
1308:P
1305:C
1300:=
1297:)
1293:T
1289:(
1284:1
1280:F
1276:=
1272:P
1247:P
1225:M
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1195:P
1164:.
1159:4
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1145:T
1111:A
1083:1
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1022:3
1017:P
1014:C
991:T
988:P
964:.
955:A
940:A
936:A
932:x
928:i
925:=
920:A
869:A
854:A
836:+
831:A
812:A
804:=
801:)
798:Z
795:(
766:)
757:A
748:,
743:A
735:(
732:=
723:Z
702:2
699:,
696:1
693:=
684:A
680:,
677:A
653:)
638:,
635:I
632:(
629:=
593:x
583:A
579:A
566:=
557:A
553:A
549:x
518:A
490:A
456:A
441:A
437:A
433:x
429:i
421:A
413:=
410:)
407:x
404:(
399:A
370:M
335:3
330:P
327:C
305:(
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286:R
254:4
249:R
225:3
220:P
217:C
188:.
183:2
178:C
154:4
149:R
95:0
92:=
87:)
84:B
66:A
63:(
54:A
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